GENERALISATION OF BOREL'S RELATION FOR ENTIRE DIRICHLET SERIES WITH ARBITRARY POSITIVE EXPONENTS

Abstract

For a entire Dirichlet series $F(z)=\sum_{n=0}^{+\infty} a_n e^{z\lambda_n}$  with arbitrary unbounded sequence of exponents$\{ \lambda_n : n\ge 0\}\subset \mathbb{R}_+$, conditions for $\omega(\ln M(x,F))-\omega(\ln\mu(x,F))\to 0$ as$x\to+\infty$ $(x\notin E,$ $\int_E d\ln x< +\infty),$ where 
$M(x,F) =\sup\{|F(x+iy)|: y\in\mathbb{R}\},$ $\mu(x,F) = \max \{|a_n| e^{x\lambda_n}: n \ge 0\}$ $(x\in\mathbb{R})$ are established.